On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction
The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a poly- nomial solution. They have shown that for μ ≠ 0 this happens ex- actly when l ∈ N and the parameters (λ, μ) lie on an algebraic curve Γl ⊂ C2(λ,μ) called the l-spectral curve and defined as zero locus of de- terminant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dy- namical systems on 2-torus depending on real parameters (B,A;ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in R2(B,A) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every l ∈ N. We also calculate its genus for l ⩽ 20 and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in R3 . We show that, unexpectedly, its complexification is(B,A,ω−1) a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.